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| Mirrors > Home > MPE Home > Th. List > raleqbidvv | Structured version Visualization version GIF version | ||
| Description: Version of raleqbidv 3346 with additional disjoint variable conditions, not requiring ax-8 2110 nor df-clel 2816. (Contributed by BJ, 22-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| raleqbidvv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| raleqbidvv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| raleqbidvv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | raleqbidvv.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | raleqbidvv.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 3 | 2 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | 
| 4 | 1, 3 | raleqbidva 3332 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ral 3062 df-rex 3071 | 
| This theorem is referenced by: rexeqbidvvOLD 3337 raleqbi1dv 3338 gpg5nbgrvtx03star 48036 postcposALT 49172 | 
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